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2 years ago

What is the reason for statement 7 in the given proof?

Mathematics

1 answer:

3241004551 [841]2 years ago

5 0

Answer: definition of slope

Step-by-step explanation:

The results in step 7 come from substituting the coordinates into the slope formula.

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Answer:

see explanation

Step-by-step explanation:

Using the chain rule

Given

y = f(g(x)), then

$\frac{dy}{dx}$ = f'(g(x)) × g'(x) ← chain rule

and the standard derivatives

$\frac{d}{dx}$ ( $log_{a}$ x ) = $\frac{1}{xlna}$ , $\frac{d}{dx}$ (lnx) = $\frac{1}{x}$

(a)

Given

y = $log_{a}$ $\sqrt{(1+x)}$

$\frac{dy}{dx}$ = $\frac{1}{lna\sqrt{(1+x)} }$ × $\frac{d}{dx}$ ( $(1+x)^{\frac{1}{2} }$

= $\frac{1}{lna\sqrt{(1+x)} }$ × $\frac{1}{2}$ $(1+x)^{-\frac{1}{2} }$ × $\frac{d}{dx}$ (1 + x)

= $\frac{1}{lna\sqrt{(1+x)} }$ × $\frac{1}{2\sqrt{(1+x)} }$ × 1

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= $\frac{1}{(1+x)lna^2}$

(b)

Given

y = ln sinx

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= $\frac{1}{sinx}$ × cosx

= $\frac{cosx}{sinx}$

= cotx

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